A215267 Decimal expansion of 90/Pi^4.

9, 2, 3, 9, 3, 8, 4, 0, 2, 9, 2, 1, 5, 9, 0, 1, 6, 7, 0, 2, 3, 7, 5, 0, 4, 9, 4, 0, 4, 0, 6, 8, 2, 4, 7, 2, 7, 6, 4, 5, 0, 2, 1, 6, 6, 8, 2, 7, 4, 4, 3, 6, 4, 4, 6, 3, 5, 1, 2, 3, 1, 9, 2, 4, 7, 7, 6, 2, 9, 6, 4, 0, 7, 9, 9, 6, 7, 2, 8, 2, 2, 4, 1, 6, 5, 1, 4, 3, 7, 3, 6, 5, 7, 6, 1, 4, 4, 1, 5

Offset: 0

Author

Jimmy Zotos, Aug 07 2012

Keywords

• nonn
• cons
• easy

Comments

Decimal expansion of 1/zeta(4), the inverse of A013662. This is the probability that 4 randomly chosen natural numbers are relatively prime.

Also the asymptotic probability that a random integer is 4-free. See equivalent comments in A088453, A059956. - Balarka Sen, Aug 08 2012

The probability that the greatest common divisor of two numbers selected at random is squarefree (Christopher, 1956). - Amiram Eldar, May 23 2020

Examples

			0.92393840292159016702375049404068247276450216682744364463512319...
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 231.

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
• John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
• Math Forums, Probability that a number is composite, Aug 2012.
• Index entries for transcendental numbers

Crossrefs

Cf. A013662, A046100 (4-free numbers), A059956 (1/zeta(2)).

Programs

Mathematica

RealDigits[90/Pi^4, 10, 100][[1]] (* Bruno Berselli, Aug 07 2012 *)

Maxima

90/%pi^4; /* Balarka Sen, Aug 08 2012 */

PARI

90/Pi^4 \\ Charles R Greathouse IV, Aug 07 2012

Formula

Reciprocal of A013662.

1/zeta(4) = 90/Pi^4 = Product_{k>=1} (1 - 1/prime(k)^4) = Sum_{n>=1} mu(n)/n^4, a Dirichlet series for the Möbius function mu. See the examples in Apostol, here for s = 4. - Wolfdieter Lang, Aug 07 2019